Large-sample convergence conjecture for infinitesimal gradient boosting

Let $(\hat{F}_t)_{t\geq 0}$ be the infinitesimal gradient boosting trajectory, let $\beta\geq 0$ be the tree-randomization parameter, let $\mathcal{P}_d(F^*)$ denote the projection of the regression target onto the relevant degree-$d$ function space, let $L$ be the loss, and let $\mathbb{W}^\infty\cap L^2_d$ be the function space containing the boosting trajectory. Large-sample convergence conjecture. (i) In regression, when $\beta>0$, strong convergence holds

$$\hat{F}_t \overset{L^2}{\longrightarrow} \mathcal{P}_d(F^*),\quad\text{as }t\to\infty.$$

(ii) In the general case, when $\beta\geq 0$,

$$\mathbb{E}[L(Y,\hat{F}_t(X))]\longrightarrow \inf_{F\in\mathbb{W}^\infty\cap L^2_d}\mathbb{E}[L(Y,F(X))],\quad\text{as }t\to\infty.$$

The preceding proposition establishes only weak convergence in regression in general, with strong convergence proved for completely random trees ($\beta=0$); the conjecture asserts the stronger result for $\beta>0$ and extends the expected asymptotic optimality to classification and other losses. The paper does not prove these claims.